AUTOMATED MOUNTING BIAS CALIBRATION for AIRBORNE LIDAR SYSTEM
Jing Zhang a, Wanshou Jiang a, *, San Jiang a
a LIESMARS, Wuhan University, 129 Luoyu Road, Wuhan, China
[email protected], [email protected], [email protected]
Commission IV, WG IV/4
KEYWORDS: LIDAR, calibration, mounting bias, bore-sight angle, lever-arm offset, Virtual Corresponding Point Model
ABSTRACT:
Mounting bias is the major error source of Airborne LIDAR system. In this paper, an automated calibration method for estimating LIDAR system mounting parameters is introduced. LIDAR direct geo-referencing model is used to calculate systematic errors. Due to LIDAR footprints discretely sampled, the real corresponding laser points are hardly existence among different strips. The traditional corresponding point methodology does not seem to apply to LIDAR strip registration. We proposed a Virtual Corresponding Point Model to resolve the corresponding problem among discrete laser points. Each VCPM contains a corresponding point and three real laser footprints. Two rules are defined to calculate tie point coordinate from real laser footprints. The Scale Invariant Feature Transform (SIFT) is used to extract corresponding points in LIDAR strips, and the automatic flow of LIDAR system calibration based on VCPM is detailed described. The practical examples illustrate the feasibility and effectiveness of the proposed calibration method.
1. INTRODUCTION
Airborne LIDAR is a very effective and accurate technology for 3D information collection, and it is widely adopted for land mapping, resource exploration, and disaster monitoring, 3D urban reconstruction and so on. A typical LIDAR system consists of three main components, a differential Global Position System (DGPS) to provide position information, an Inertial Measurement Unit (IMU) for attitude determination, and a laser scanning system to provide range and angle information (Baltsavias, 1999). As a multi-sensor integrated system, the positioning accuracy of individual laser footprint is affected by many factors. The main limiting factors contain the errors of the platform position and orientation derived from the DGPS/INS data and assembly errors related to misalign among sensors. The research of LIDAR data adjustment can eliminate systematic errors effectively and improve positioning accuracy. It is very important for LIDAR applications. Nowadays, existing methods can be divided into two categories: one is data-driven approach (often referred to as strip adjustment), the other is model-driven approach (often referred to as system calibration) (Shan, 2008).
LIDAR strip adjustment methods have been developed in the past few years. For the sake of mitigating the impact of systematic discrepancies between overlapping strips, various mathematic models are employed to approximately express LIDAR strip deformation (Kilian, 1996; Crombaghs, 2000; Maas, 2002; Vosselman, 2004; Filin, 2004). As the LIDAR system errors cause complicated discrepancies between strips, it is impossible to find an optimal generalized model satisfied all sorts of strip deformation. Actually, the model selection is mostly empirical. The adjustment precision depends on the appropriateness of model selection.
Alternatively, system calibration methods use rigorous physical model to calibrate LIDAR systematic biases. Then discrepancies between LIDAR overlapping strips can be reduced by compensating systematic errors directly. Schenk
(2001) analyzed the LIDAR systematic errors, and introduced a manual calibration method to solve each parameter individually. Due to correlation of systematic errors, most methods adopt partial parameters for calibration. For example, Morin (2002) solves the bore-sighting angles and the scanner torsion simultaneously. Skaloud (2006) adopts bore-sighting angles and laser range bias. Habib (2007) selects bore-sighting angles and lever-arm components. Compared with strip adjustment, system calibration utilized laser point raw measurements (such as ranges, mirror angles, position and orientation information for each pulse) and solved system biases directly. It is more reliable and stable.
LIDAR adjustment process relies on the identification of corresponding primitives for overlapping LIDAR strips. There are three types of corresponding primitives: point primitive, linear primitive and planar primitive. Considering discrete distribution of laser footprint, identifying corresponding point among LIDAR strips is almost impossible. Currently, most methods choose linear and planar primitives (Skaloud, 2006; Filin, 2006; Habib, 2007), and determine corresponding lines and planars manually. To ensure the accurate calibration results, these methods require lines in different directions, planars of different slope and aspect (Habib, 2010), which are difficult available in practice.
3, real dataset from Anyang, China is used to test our method. The results show the feasibility and efficiency of the proposed method.
2. METHOD
2.1 Direct Geo-referencing Model
According to LIDAR position equation (Baltsavias, 1999; Schenk, 2001), space coordinate of each laser footprint is calculated from system raw measurements. In various kinds of systemic errors, we choose bore-sighting angles and lever-arm offsets (which are known as the mounting bias) as system calibration parameters. Bore-sight angles describe the rotate angles between IMU body frame to laser scanner frame, and lever-arm offsets are position shift between GPS antenna center to laser scanner center. Compared with other systematic error sources, mounting bias is easy to change. However, each time the LIDAR system is installed, re-installed, or moved within the aircraft, small changes in the alignment may occur. So LIDAR mounting biases should be calibrated frequently to obtain high quality laser scanner data (Morin, 2002; Habib, 2008).
Up to now, differential GPS technique has demonstrated positioning accuracy at the 3-5 cm level. Considering only mounting bias, LIDAR position equation can be simplified as equation(1):
X Y Z is coordinates of laser footprints in the
reference frame; , , T gps gps gps
X Y Z
⎡ ⎤
⎣ ⎦ is coordinates of GPS
antenna center;
R
imuis the orientation matrix from IMU body frame to reference frame parameterized by roll (r), pitch (p)and heading (h) angle observations measured by IMU unit;
misalign
R is defined by bore-sight angles, which designates the
rotation matrix between IMU body frame and laser scanner frame. The bore-sight angles is parameterized by
[
α β γ, ,]
;scan
R
is the rotation matrix defined by laser scanner rotation angleθ
;ρ
is range measurement from laser range-finder;, , T
offset offset offset
X Y Z
⎡ ⎤
⎣ ⎦ defines the lever-arm offsets from laser scanner center to GPS antenna center.
In equation(1), we designate lever-arm offset
, , T
offset offset offset
X Y Z
⎡ ⎤
⎣ ⎦ and bore-sight angles
[
α β γ
, ,]
are unknown coefficients, all the other variants are known values which can obtain from laser point raw measurements. Observation equation of unknown coefficients is given in formula(2):L is coordinate expectations of
footprints; , , , , , T
parameters of mounting parameters. Coordinate expectations can gain from ground control points, or adopt mean value of conjugate points just as self-calibration. According to formula(2), one laser point can formulate three equations. So selecting two more points, the least square solution of mounting
parameters can be obtained.
Indirect adjustment model is used to solve mounting parameters, which is shown in formula(3). Wherel= −L (AX0+Δ)= −L L0;
A is the coefficient matrix of unknown parameters; L0 is observation approximation, and X0
is unknown parameters approximation, x is unknown parameters correction.
= −
V A x l (3) The solution is derived by least square adjustment in formula(4).
P is weight matrix of observation. If all observations are of the same importance, P could be identity matrix.
1
( T )− T
=
x A PA A Pl (4)
2.2 Virtual Corresponding Point Model
As LIDAR points are sampled discretely, real conjugated laser footprints are hardly existence. We proposed a novel method which termed Virtual Corresponding Point Model (VCPM) to resolve this corresponding problem among discrete laser points. As shown in Fig. 1, supposing point p and q are corresponding
points in different strips. Point p is not a real laser footprint,
three real footprints p1, p2 and p3 in its nearest neighborhood
are selected to construct a triangle enclosing p. We integrate p
and p1, p2, p3 into VCPM P. In the same way, VCPM Q can be
constructed for q. Now two rules are defined for VCPM to
calculate the horizontal and vertical coordinates of corresponding points by three laser footprints.
p
Fig. 1 Virtual Corresponding Point Model Rule 1: Coplanar constraint
When p and three relevant footprints, p1, p2, p3, are coplanar, the vertical coordinate of p can be determined by linear interpolation. Let coordinates of point p1, p2, p3
are
(
)
X Y . The vertical coordinate of point p
is determined by equation(5):
2 1
Rule 2: Rigid-body assumption
According to equation(1), the modification of mounting parameters will vary footprints coordinates, furthermore affect corresponding point coordinates. Because p and p1, p2, p3
locate in a small area, when footprint coordinates varying, assuming the entity deformation of VCPM P obeys rigid-body transformation. After mounting parameters modified, coordinates of corresponding point p could be recalculated by
Where 0, ,0 0 T parameters changed; Rrigid is the rotation matrix and
[
, ,]
Tx y z
Δ Δ Δ is offset vector of rigid transformation. The rotation matrix and offset vector could be calculated from three real footprints. Whenever mounting parameters change, coordinates of three footprints before and after varying are used to solve rigid transformation parameters. Then using previous corresponding point coordinates and transformation parameters, current coordinates of corresponding point can be determined. Utilizing two rules above, we could recalculate and update corresponding point coordinate using real footprints when systematic parameters varying. VCPM provides an approach to solve point corresponding in discrete LIDAR points.
2.3 Adjustment Process
Based on VCPM, the flowchart of LIDAR mounting bias adjust is shown in Fig. 2. The details of each step are as follows:
Fig. 2 Workflow of calibration Step 1Extract corresponding points
Corresponding points identification is the base of LIDAR system calibration. Corresponding points can be determined manually. In this paper, a full automatic approach is given. The details are introduced in chapter 2.4.
Step 2Construct VCPM
For each corresponding point, all real footprints within a given radius (e.g. 2 meters, depend on average point spacing) are extracted. Then, a delaunay triangular network (TIN) would be constructed from these footprints. Finally, pick up the triangle encompasses the corresponding point and take triangle vertexes into VCPM. If laser footprints within specified radius are less than 3, TIN construction is failed. In this case, this corresponding point will be deleted.
Step 3Initialize mounting parameters
Initialize values for all mounting parameters. These parameters can be initialized by previous calibration. In practical, mounting bias are usually very small, we can initialize all as zero.
Step 4Calculate corresponding points coordinates
Corresponding points coordinates are calculated based on VCPM. For point k, horizontal coordinates
(
,)
k k X Y are determined in extraction step. If k and surrounding laser
footprint is coplanar, the vertical coordinate k
Z could be interpolated according formula(5). In the iteration process of adjustment, mounting parameters will vary. Current corresponding point coordinates can be calculated by formula(6).
Step 5Calculate Observation value
Calculate expectation value of corresponding points coordinates and constitute observation vector (L in
formula(2)). Suppose corresponding point k visible in n strips, expectation values are calculated as(7).
1
X is coordinate expectation;
i
k
X is coordinate in
strip i.
Step 6Calculate Coefficient Matrix
Due to the complex relationship between observations and unknowns, it’s difficult to linearize the functional model. We adopt finite difference method to calculate the coefficient matrix of unknowns. Calculation is as follows:
( , )
For equation(8), x, y are unknowns, partial derivatives can be calculated in formula(9). Coefficient matrix of mounting parameters is formulated as(10):
e e e e e e
Step 7Solve mounting parameters
Mounting parameters are solved by least square. It is a classical parameters minimization problem, among various solutions, we choose LM algorithm (MORÉ, 1978).
Step 8Iteration Condition
Once residual error of corresponding points is less than a threshold or calibration parameters are converged, stop the iteration and output the adjustment results. Otherwise, go to Step 4.
2.4 Automatic Corresponding Point Extraction
VCPM provides an approach to solve point corresponding in discrete LIDAR points. So many sophisticated points matching method in traditional photogrammetry can be employed to extract corresponding points in LIDAR data. In this section, automatic corresponding point extraction is introduced in 5 steps.
(1) Interpolation
(2) Matching by SIFT
SIFT features (Lowe, 2004) are extracted from each interpolated LiDAR strip. Then the features are matched between pair of strips.
(3) RANSAC Estimation
Since there might be some mismatches in SIFT matching, RANSAC estimation (Fischler, 1981) is used to select correct matches. We use affine transformation as a geometrical constraint in this step.
(4) Tracking
With a set of geometrically consistent matches between each strip pair, a tracking procedure is employed to connect corresponding points across multiple strips.
(5) Classification of corresponding points
Vertical coordinates of corresponding points are determined by linear interpolation. When corresponding point and surrounding laser footprints are not coplanar, the vertical coordinate is unreliable. Surface roughness is adopted to analysis and classify all accepted corresponding points (Shepard, 2001). Take each corresponding point as center, statistics RMSE (Root Mean Square Error) of laser footprints elevation within a specified radius (e.g. 2m), and use this value measure surface roughness. When corresponding points lie on vegetation or building edges the value of surface roughness is relatively big. When corresponding points lie on steady planes, the roughness is relatively small. So we compare the surface roughness with a given threshold (0.25) to classify the corresponding points into two types: horizontal-vertical points and horizontal points. In the process of adjustment, for horizontal–vertical points, x, y, z equations are used; for horizontal points, only x, y equations are used.
3. EXPERIMENT AND ANALYSIS
3.1 Experiment Description
To test the proposed method, real LIDAR dataset is used for experiment. Experimental area is in urban area located at Anyang, Henan province, China. The terrain is relatively flat, and plenty of buildings distribute in this area. LIDAR dataset is acquired with ALS50II. The whole dataset contains 8 LIDAR strips (shown in Fig. 3), and details of each strip is list in Tab. 1.
a. 1~4 strips b. 5~8 strips Fig. 3 LIDAR strips in experimental area
Tab. 1 Essential information of 8 strips
8 strips can be divided into two groups. Strips 1-4 are four cross
strips; the overlap of the same direction strips is 100 percent. Strips 5-8 are four parallel strips; the maximal overlap between strips is 15 percent. In order to inspect experimental result, we assign a different color for each strip (Tab. 1).
The experiment is composed of two steps. Firstly, Strips 1-4 are used to confirm the feasibility of the proposed method. Then the calibration results of mounting bias are used to correct remaining 4 strips. Quantitative analysis of relative accuracy after correction is given to confirm the efficiency of the proposed method.
3.2 Results Analysis
3.2.1 Experiment 1: In this experiment, strips 1-4 are used to solve LIDAR system mounting parameters. Firstly, SIFT features are extracted form interpolated LIDAR intensity image. The number of original feature points of each strip is shown in. Then SIFT features are matched for each pair of strips, the initial pairwise matches are shown in Tab. 2. The value in row i, column j is the match number between image pair (i, j). After RANSAC estimation, geometrically consistent matches are shown in Tab. 3. Finally, after tracking and classification, the number of received corresponding points is shown in Tab. 4. The distribution of corresponding points is shown in Fig. 4.
Strip 1 2 3 4
Count 12813 17909 30979 33220
Tab. 2 Original SIFT feature points
Strip 1 2 3 4
1 0 31 717 90
2 - 0 80 683 3 - - 0 245 4 - - - 0
Tab. 3 Initial pairwise matches
Strip 1 2 3 4
1 0 13 289 36
2 - 0 35 301 3 - - 0 89 4 - - - 0
Tab. 4 Matches of RANSAC estimation
Horizontal – Vertical 312
Horizontal 426
Total Num 738
Tab. 5 Received tie points
Fig. 4 Distribution of final tie points
The solved mounting parameters are listed in Tab. 6. The RMSE
Strip
ID Direction
Average Flight Height
Average point
Spacing Colour
of corresponding points pre- and post- adjustment is shown in Tab. 7. After mounting bias correction, that RMSE of corresponding points is great improved.
Xoffset Yoffset Zoffset
Level-arm(meter)
4.0768-e5 5.41939-e4 1.67461-e4
α β γ
Bore-sight(radian)
-5.13916-e3 2.38587-e3 -7.00623-e4
Tab. 6 Mounting parameters
X(m) Y(m) Z(m) Pre 3.577878 3.752160 0.834701 Post 0.149332 0.142443 0.036839
Tab. 7 RMSE of tie points
We select several profiles in overlapping region to check the offsets among strips. Fig. 5a shows a certain profile of original LIDAR strips, there are obvious offsets between gable roof and ground. Fig. 5 shows the same profile of LIDAR strips after adjustment, roof and ground are coincided very well. To check the horizontal accuracy between strips, LIDAR points are rendered by intensity. Fig. 6 shows the overlay drawing of strip 1-4. Mounting bias causes significant horizontal displacement (Fig. 6a). After adjustment, the runway center line is accurately registered (Fig. 6b).
a. Before adjustment
b. After adjustment Fig. 5 Contrast of profile
a. Before adjustment b. After adjustment Fig. 6 Contrast of LIDAR Intensity
3.2.2 Experiment 2: In this experiment, mounting parameters acquired from experiment 1 are applied to corrected strips 5-8 directly. Elevation profiles of each strip pair (5 – 6, 6 – 7, 7 – 8) are selected for visual inspection. The color of laser points are specified by strip number, color scheme is in Tab. 1. As shown in Fig. 7, a large misalignment exists between raw strips. After correction by mounting parameters, the misalignment is almost eliminated, and the relative accuracy between strips is greatly improved.
a. pre-corrected of strip 5 ~ 6
b. post-corrected of strip 5 ~ 6
c. pre- corrected of strip 6 ~ 7
d. post- corrected of strip 6 ~ 7
e. pre- corrected of strip 7 ~ 8
f. post- corrected of strip 7 ~ 8 Fig. 7 Contrast of profiles
Fig. 8 Hypsometric layer map of overlapping region D-DSM
a. High buildings b. Low buildings Fig. 9 Profiles of buildings
4. CONCLUSIONS
LIDAR system bias calibration can improve data accuracy effectively. It is very important for LIDAR applications. Prior methods exist more or less shortages in feasibility, accuracy and automaticity. This paper introduces an automated method for airborne LIDAR mounting biases calibration. The calibration process uses the LIDAR direct geo-referencing model to solve the level-arm offsets and bore-sighting angles. As LIDAR points are discrete sampling, real conjugated points don’t exist between LIDAR strips; Virtual Corresponding Point Model (VCPM) is used as corresponding primitive. Using the given rules, virtual corresponding points easily associate with real laser footprints. Then the adjustment based on corresponding points is applied directly. Automatic flow is designed to determine corresponding points between LIDAR strips, which contains SIFT matching, RANSAC estimation and classification.
A set of real LIDAR strips is used to test proposed method. The experiment is divided into two steps. Firstly, 4 LIDAR strips are used to solve system mounting parameters with our method. Then the remainder 4 strips are corrected by acquired mounting parameters and the relative accuracy is analyzed to verify the efficiency of calibrated parameters. The results show the feasibility and efficiency of the proposed procedure.
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